Download Fuzzy strongly g*- Super closed Sets

IJREAS
VOLUME 6, ISSUE 5 (May, 2016)
(ISSN 2249-3905)
International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR – 6.573)
Fuzzy strongly g*- Super closed Sets
M.K. Mishra1,
M.Shukla2
Professor EGS PEC Nagapattinam,
Asst. Prof Arignar Anna College Karaikal
Abstract:
In this paper, we introduce and investigate the concept of fuzzy strongly generalized g* - Super
closed sets (briefly fuzzy strongly g*-Super closed set) and investigate the relation between the
associated fuzzy topology.
Keywords: fuzzy super closure, fuzzy super interior, fuzzy super open set, fuzzy super closed set
g*-Super closed set, Fuzzy Strongly g*-super closed set
1. Preliminaries
Let X be a non empty set and I= [0,1]. A fuzzy set on X is a mapping from X in to I. The null fuzzy set
0 is the mapping from X in to I which assumes only the value is 0 and whole fuzzy sets 1 is a
mapping from X on to I which takes the values 1 only. The union (resp. intersection) of a family {Aα:
Λ}of fuzzy sets of X is defined by to be the mapping sup A α (resp. inf Aα) . A fuzzy set A of X is
contained in a fuzzy set B of X if A(x) ≤ B(x) for each xX. A fuzzy point xβ in X is a fuzzy set defined
by xβ (y) = β for y=x and x(y) =0 for y  x, β[0,1] and y  X .A fuzzy point xβ is said to be quasicoincident with the fuzzy set A denoted by xβqA if and only if β + A(x) > 1. A fuzzy set A is quasi –
coincident with a fuzzy set B denoted by AqB if and only if there exists a point xX such that A(x) +
B(x) > 1 .A ≤ B if and only if (AqBc).
A family  of fuzzy sets of X is called a fuzzy topology [2] on X if 0,1 belongs to  and  is
super closed with respect to arbitrary union and finite intersection .The members of  are called
fuzzy super open sets and their complement are fuzzy super closed sets. For any fuzzy set A of X the
closure of A (denoted by cl(A)) is the intersection of all the fuzzy super closed super sets of A and
the interior of A (denoted by int(A) )is the union of all fuzzy super open subsets of A.
Defination1.1:A subset A of a fuzzy topological space (X,  ) is called
1. Fuzzy Super closure scl(A)={xX:cl(U)A≠}
2. Fuzzy Super interior sint(A) ={xX:cl(U)≤A≠}
3. Fuzzy super closed if scl(A )  A.
4.
Fuzzy super open if 1-A is fuzzy super closed sint(A)=A
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5. Fuzzy pre super open set if A≤ int(cl(A)) and fuzzy pre-Super closed set if cl(int(A))≤ A.
6. Fuzzy semi super open set if A≤cl(int(A)) and fuzzy semi super closed set if int(cl(A)) ≤A.
7. Fuzzy -Super open set if A≤ int(cl(int(A))) and an fuzzy -Super closed set if cl(int(cl(A)))
≤A.
8. Fuzzy semi pre super open set (fuzzy - A subset A of a fuzzy topological space (X,) is called
super open set) if A ≤ cl(int(cl(A))) and fuzzy semi-pre super closed set if int(cl(int(A))) ≤A.
9. fuzzy g- super closed if cl(A) ≤ G whenever A ≤ G and G is super open.
10. fuzzy g- super open if its complement 1-A is fuzzy g- super closed.
11. Fuzzy - super closed set if A = cl(A) where cl(A) = {x  X : int(cl(U))  A ,U  and x 
U} .
12. Fuzzy - super closed set if A =cl(A) where cl(A) = {x  X : (cl(U))  A , U  and x  U }.
13. Fuzzy g-super closed if cl(A) ≤ U , whenever A≤ U and U is fuzzy super open in (X,).
14. Fuzzy semi generalized super closed set (briefly fuzzy sg-super closed) if scl(A) ≤ U ,
whenever A ≤ U, U is fuzzy semi super open in (X,  ).
15. Fuzzy generalized semi-super closed set (briefly gs-super closed) if scl(A)≤U whenever A ≤
U , U is fuzzy super open in (X,  ).
16. Fuzzy generalized  -super closed (briefly fuzzy g -super closed) if cl(A)≤U whenever A ≤
U and U is fuzzy -Super open in (X,).
17. Fuzzy  generalized super closed set (briefly fuzzy g-super closed) if cl(A) ≤ U whenever
A ≤ U and U is fuzzy super open in (X,  ).
18. Fuzzy generalized semi pre-super closed set (briefly fuzzy gsp-super closed) if spcl(A) ≤ U
whenever A ≤ U and fuzzy U is super open in (X,  ).
19. Fuzzy regular generalized super closed set (briefly fuzzy r-g-super closed) if cl(A) ≤ U
whenever A ≤ U and U is fuzzy regular super open in(X,  ).
20. Fuzzy generalized pre super closed set (briefly fuzzy gp-super closed) if pcl(A) ≤U,
whenever A ≤ U and U is fuzzy super open in (X,  ).
21. Fuzzy generalized pre regular super closed set (briefly fuzzy gpr-super closed ) if pcl(A) ≤
U, whenever A ≤ U and U is fuzzy regular Super open in (X,  ).
22. Fuzzy  generalized super closed set (briefly fuzzy g-super closed) if cl≤ U whenever A ≤ U
and U is fuzzy super open in (X,  ).
23. Fuzzy -generalized super closed set (briefly fuzzy g super closed) if cl(A)≤U whenever A
≤ U and U is fuzzy super open in (X,  ).
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24. Fuzzy g* -super closed set if cl(A) ≤ U whenever A ≤ U and U is fuzzy g-super open in (X, ).
Remark 1.1[5]:- Every fuzzy closed set is fuzzy super closed but the converses may not be true.
Remark 1.2[5]:- Let A and B are two fuzzy super closed sets in a fuzzy topological space (X,), then
A  B is fuzzy super closed.
Remark 1.3[5]:- The intersection of two fuzzy super closed sets in a fuzzy topological space (X,)
may not be fuzzy super closed.
2 Fuzzy Strongly g*-super closed sets
In this section we have introduce the concept of fuzzy strongly g*-Super closed sets in topological
space and we investigate the group of structure of the set of all fuzzy strongly g*-super closed sets.
Definition 2.1: Let (X,) be a fuzzy topological space and A be its subset , then A is fuzzy strongly
g*-super closed set if cl(int(A)) ≤ U whenever A ≤ U and U is fuzzy g-Super open.
Theorem 2.1: Every super closed set is Fuzzy strongly g*-super closed set.
Proof. The proof is immediate from the definition of super closed set.
Proof: Suppose A is fuzzy g*-super closed in X. Let G be a fuzzy super open set containing A in X.
Then G contains cl(A). Now G cl(A) cl(int(A)). Thus A is fuzzy strongly g*-super closed in X.
Theorem 2.3: If A is a subset of a fuzzy topological space X is super open and fuzzy strongly g*super closed then it is fuzzy super closed.
Proof: Suppose a subset A of X is both fuzzy super open and Fuzzy strongly g*-super closed. Now A
cl(int(A))  cl(A). Therefore A cl(A). Since cl(A) A. We have A cl(A). Thus A is fuzzy super
closed in X.
Corollary 2.1: If A is both super open and fuzzy strongly g*-super closed in X then it is both fuzzy
regular super open and fuzzy regular super closed in X.
Proof: As A is Super open A = int(A) = int(cl(A)), since A is fuzzy Super closed . Thus A is fuzzy
regular super open . Again A is fuzzy super open in X, cl(int(A))=cl(A). As A is fuzzy super closed
cl(int(A))=A. Thus A is fuzzy regular super closed.
Corollary 2.2: If A is both fuzzy Super open and Fuzzy strongly g*-super closed then it is fuzzy rgsuper closed.
Theorem 3.4: If a subset A of a fuzzy topological space X is both fuzzy strongly g*-super closed and
fuzzy semi super open then it is fuzzy g*-super closed.
Proof: Suppose A is both fuzzy strongly g*-Super closed and fuzzy semi super open in X, Let G be a
fuzzy super open set containing A. As A is fuzzy strongly g*-super closed, G  cl(int(A)). Now G 
cl(A). Since A is fuzzy semi super open. Thus A is fuzzy g*-super closed in X.
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Corollary 3.3: If A subset A of a fuzzy topological space X is both fuzzy strongly g*-super closed and
fuzzy super open then it is fuzzy g*-super closed set.
Proof: As every super open set is semi super open by the above theorem the proof follows.
Theorem 2.5: A set A is fuzzy strongly g*-super closed iff cl(int(A))−A contains no non empty fuzzy
super closed set.
Proof: Necessary: Suppose that F is non empty fuzzy super closed subset of cl(int(A)). Now
F ≤ cl(int(A)) − A implies F ≤cl(int(A))  Ac, since cl(int(A)) − A = cl(int(A))  Ac. Thus F ≤ cl(int(A)).
Now F ≤ Ac implies A ≤ Fc. Here Fc is fuzzy g-Super open and A is fuzzy strongly g*-super closed, we
have cl(int(A)) _ Fc. Thus F ≤ (cl(int(A))) c. Hence F ≤ (cl(int(A)))  (cl(int(A)))c =  Therefore F = ≤(
cl(int(A)) − A contains no non empty fuzzy super closed sets.
Sufficient: Let A ≤ G, G is fuzzy g-Super open. suppose that cl(int(A)) is not contained in G then
(cl(int(A)))c is a non empty fuzzy super closed set of cl(int(A)) −A which is a contradiction.
Therefore cl(int(A)) ≤ G and hence A is fuzzy strongly g* -super closed .
Corollary 2.4: A fuzzy strongly g*-Super closed set A is fuzzy regular super closed iff cl(int(A)) – A
is fuzzy super closed and cl(int(A)) ≤ A.
Proof: Assume A that A is fuzzy regular super closed. Since cl(int(A)) = A, cl(int(A)) −A =  is fuzzy
regular super closed and hence fuzzy super closed . Conversely assume that cl(int(A))−A is fuzzy
super closed . By the above theorem cl(int(A))−A contains no nonempty fuzzy super closed set
.Therefore cl(int(A)) − A = .Thus A is fuzzy regular super closed .
Theorem 2.6: Suppose that B ≤ A ≤ X, B is fuzzy strongly g*-super closed set relative to A and that
both fuzzy super open and fuzzy strongly g * -super closed subset of X then B is fuzzy strongly g* super closed set relative to X.
Proof: Let B ≤ G and G be a fuzzy super open set in X. But given that B ≤ A ≤ X, therefore B ≤ A and B
≤G. This implies B ≤ A  G. Since B is fuzzy strongly g*- super closed relative
to A,
cl(int(B))≤AG.(ie)Acl(int(B))≤AG.This implies A(cl(int(B)))≤G.Thus (A(cl(int(B)))) 
(cl(int(B)))c ≤ G(cl(int(B)))c implies A  (cl(int(B)))c ≤ G ((cl(int(B)))c. since A is fuzzy strongly
g*-super closed in X, we have (cl(int(A))) ≤ G  (cl(int(B)))c. Also B ≤ A  cl(int(B)) ≤ cl(int(A)).
Thus cl(int(B)) ≤ cl(int(A)) ≤ G ≤ (cl(int(B)))c. Therefore B is fuzzy strongly g* -super closed set
relative to X.
Corollary 2.5: Corollary 3.14: Let A be fuzzy strongly g* -super closed and suppose that F is fuzzy
super closed then A  F is fuzzy strongly g* -super closed set.
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Proof: To show that AF is fuzzy strongly g*-Super closed, we have to show cl(int(A\F)) ≤ G
whenever A  F ≤G and G is fuzzy g-super open. A  F is fuzzy super closed in A so it is fuzzy
strongly g* -super closed in B. By the above theorem A  F is fuzzy strongly g* -super closed in X.
Since A  F ≤ A ≤ X.
Theorem 2.7: Theorem 3.15: If A is fuzzy strongly g* Super closed and A ≤ B≤cl(int(A)) then B is
fuzzy strongly g *-super closed .
Proof: Given that B ≤cl(int(A)) then cl(int(B)) ≤ cl(int(A)), cl(int(B)) −B ≤ cl(int(A)) − A .Since A ≤ B.
As A is fuzzy Strongly g* -super closed by the above theorem cl(int(A)) − A contains no non empty
fuzzy super closed set, cl(int(B)) – B contains no empty fuzzy super closed set. Again by theorem
3.13, B is fuzzy strongly g*-super closed set.
Theorem 2.8: Theorem 3.16: Let A ≤ Y ≤ X and suppose that A is fuzzy strongly g* -super closed in
X then A is fuzzy strongly g* -super closed relative to Y.
Proof: Given that A ≤ Y ≤ X and A is fuzzy strongly g*-super closed in X. To show that A is fuzzy
strongly g*−super closed relative to Y, let A ≤ Y  G, where G is fuzzy g-super open in X. Since A is
fuzzy strongly g*-super closed in X,A ≤ G implies cl(int(A))≤G. (ie) Y  cl(int(A)) ≤ Y  G, where Y
 cl(int(A)) is closure of interior of A in Y. Thus A is fuzzy strongly g* -super closed relative to Y.
Theorem 2.9: If a subset A of a fuzzy topological space X is gsp-super closed then it is fuzzy
strongly g*-super closed but not conversely.
Proof: Suppose that A is fuzzy gsp- super closed set in X, let G be fuzzy super open set containing A.
Then G spcl(A),A  G  A ((int(cl(int(A)))) which implies G  int(cl(int(A))) as G is fuzzy super
open. (ie)G  cl(int(A)) − A is fuzzy strongly g* -super closed set in X.
Theorem 2.10: Theorem 3.19: Every fuzzy − super closed set is a fuzzy strongly g*-super closed
set.
Proof: The Proof of the theorem is immediate from the definition.
Theorem 2.11: Every fuzzy  -super closed set is a fuzzy strongly g*-super closed set.
Proof: The Proof of the theorem is immediate from the definition.
Theorem 2.12: Every fuzzy strongly g*-super closed set in an fuzzy  g-super closed set and
hence fuzzy gs -super closed , fuzzy gsp-super closed , fuzzy gp-super closed , fuzzy gpr -super
closed set and fuzzy rg -super closed set but not conversely.
Proof: Let A be a fuzzy strongly g*-super closed set of (X,). By above theorem, A is fuzzy g-super
closed. A is fuzzy g-super closed. We know that every fuzzy g-super closed set is fuzzy gs-super
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closed, fuzzy gsp-super closed, fuzzy gp-super closed, fuzzy gpr-super closed and fuzzy rg-super
closed. By above theorem every fuzzy strongly g*-super closed set is fuzzy gs-super closed, fuzzy
gsp -super closed and fuzzy rg -super closed.
Remark 2.1: The following are the implications of fuzzy strongly g*-Super closed set.
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